If and are two non-void relations on a set . Then, which of the following statements is incorrect
and are transitive implies is transitive
Checking for incorrect statement:
Step 1: Check for the condition if and are symmetric, then its union is symmetric or not.
A relation is reflexive if each element is related to itself.
A relation is symmetric if any one element is related to any other element, then the second element is related to the first.
A relation is transitive if any one element is related to a second and that second element is related to a third, then the first element is related to the third.
Let us consider two non-void relations on a set be:
Here, both and are symmetric, Then the union of and is given by
Which implies that is symmetric.
Step 2: Check for the condition if and are reflexive, then its intersection is reflexive or not.
Let us consider two non-void relations on a set be:
Here, both and are reflexive, Then the union of and is given by
which implies that is reflexive.
Step 3: Check for the condition if and are transitive, then its union and intersection are transitive or not.
Let us consider two non-void relations on a set be:
Here both and are transitive, Then the union of and is given by
Then, is transitive.
Again check for the intersection of and is given by ,
Therefore, is not transitive.
Hence, option (B) is correct.